NUMBER SYSTEMs
1) NATURAL NUMBERS
[N]:{1,2,3,------}
Natural numbers starts with “1” and end with ∞ [infinity] or endless
Ø In the prehistoric days the people
doesn’t know the”0” and fraction etc. The numbers 1 to 9 are used for counting
only.
2) WHOLE
NUMBERS (W)={0,1,2,3-----}
It starts from 0 with ∞
i.e endless.
*The Indians found “0” in “876” BC.
Then “0” was joined to the natural numbers became whole numbers.
W=NU{0}
Set of natural numbers is sub-set of whole numbers.
3) ZERO NUMBES (Z) ={-∞---------5,-4,-3,-2,-1,0,1,2,3-------∞)
The number system having "+”ve and “-“ve, zero and end with -∞ and end with +∞ i.e endless on both sides [The symbol ∞ express infinite].The system identifying the -ve numbers as barrows, debts, opposite directions in geometry.
E.g :If Height from earth = -6m i.e depth from earth=6m while solving a mathematical problem in heights and distances we doesn’t aware the result whether it is +v or -Ve . The result can be identified as depth if it is –ve number.
Loss should be as in "-ve“.
In zero numbering system there are no fractions, rational numbers. They are integers only.
The zero numbers set consists of natural numbers whole numbers
The number system having "+”ve and “-“ve, zero and end with -∞ and end with +∞ i.e endless on both sides [The symbol ∞ express infinite].The system identifying the -ve numbers as barrows, debts, opposite directions in geometry.
E.g :If Height from earth = -6m i.e depth from earth=6m while solving a mathematical problem in heights and distances we doesn’t aware the result whether it is +v or -Ve . The result can be identified as depth if it is –ve number.
Loss should be as in "-ve“.
In zero numbering system there are no fractions, rational numbers. They are integers only.
The zero numbers set consists of natural numbers whole numbers
N⊂W⊂Z
Set of natural numbers is sub set of
set of whole numbers
Set of whole numbers is sub set of
set of zero numbers.
3) Rational numbers
[Q]
The set of numbers having numbers of fractional between any two integers and can be written as p/q form [can be written as decimal fractions also like 7.235719].
The set of numbers having numbers of fractional between any two integers and can be written as p/q form [can be written as decimal fractions also like 7.235719].
The number of fraction be found In-between even two integer
as numbering scale is “∞” .This is called as density property
*[N⊂W⊂Z⊂Q]
The rational numbers between any two rational numbers
E.g: The rational numbers between 0 to 1
E.g: The rational numbers between 0 to 1
If the 0 to 1 scale
divided into 10 small parts
0---1/10---2/10---3/10---4/10---5/10---6/10---7/10---8/10---9/10---10/10 [i.e is 1]
If the sale is divided in to small points as 100
0--1/100---2/100---3/100---4/100---5/100--------------------------98/100---99/100---100/100 i.e 1If the sale is divided in to small points as 100
Similarly
the scale between 0&1 can be divided into “n” numbers as small. So we identify the number of rational numbers between any two rational numbers is infinite. This property is called as density property.
Irrational
numbers [Q1]: √2, √3,
√5, √6, √7, √11 etc.
The numbers,
whose value with none repeated endless decimal fractions.
Eg: √3=1.73205080757…….
Eg: √2=1.414213562373…..
Rationalization Factor: If product of two irrational numbers is a rational number. Then the two irrational numbers are called as rationalization factors each other.
E.g: √3X[-√3]=-3,-3 is rational number. So √3 is rationalization factor of -√3
E.g: [√5-√3]X[√5+√3]=[√5]2-[√3]2=5-3=2 identified as[a+b]X[a-b]=a2-b2
So rationalization factor of [√5-√3] is [√5-√3] vice versa.
Rationalization Factor: If product of two irrational numbers is a rational number. Then the two irrational numbers are called as rationalization factors each other.
E.g: √3X[-√3]=-3,-3 is rational number. So √3 is rationalization factor of -√3
E.g: [√5-√3]X[√5+√3]=[√5]2-[√3]2=5-3=2 identified as[a+b]X[a-b]=a2-b2
So rationalization factor of [√5-√3] is [√5-√3] vice versa.
Real
Numbers [R]
The
combination or unity of Rational numbers and irrational numbers is the set of
real numbers.
N⊂W⊂Q⊂R
3) Imaginary
Numbers [i]: The set of numbers are nth root of “-ve“ numbers when n is even
number.
i.e √-2, √-3,
√-9, √-7, √-25 etc.
4]
Complex numbers [C]:
The numbers,
which gives result of addition, subtraction, multiplication and division
between imaginary numbers and real numbers. i.e in the form of or a+√-b=a+ ix where 
=I, √b= x the value of b is +ve
=I, √b= x the value of b is +ve
a-√-b=a+ ix
a√-b= iax
*N⊂W⊂Q⊂R⊂C
Sub-Classification
of the integers.
1] Even
numbers: the numbers divisible by 2 are called even numbers.
E.g: 2,4, 6,
8, …….
2] odd
numbers :numbers not divisible by 2 are called odd numbers.
E.g: 1, 3,
5, 7, …….
3] Composite
numbers: the numbers having more than two factors is called composite
numbers.
E.g1] :
4 factors of 4 are 1, 2 and 4. The
number of factors is 3
E.g2]:
6 factors of 6 are 1, 2, 3 and 6 The
number of factors is 4
The numbers consist of only two factors i.e 1 and itself are
called prime numbers.
e.g1]: 2
factors of 2 are 1 and 2. The number of factors is 2
*important note: 2 is only one even prime number.
E.g 2]: 3,
7, 11, 13, 17, 19, 23, ……..
*
Important note : All
prime numbers are odd numbers but all odd numbers are not prime numbers.
Set of prime
numbers⊂set of odd numbers
Twin
prime numbers
Any pair of prime numbers has difference as 2 is called twin prime numbers.
Any pair of prime numbers has difference as 2 is called twin prime numbers.
e.g1] 3,5 difference is 5-3=2
Hence 3, 5 are twin prime numbers and 5, 7 are twin prime numbers.
Fractions:all rational numbers except complete integers are called fractions
these are in the form of p/q.further the numbers are classified as below
a]proper fractions
b]improper fractions
c]mixed fractions
a]proper fractions:These numbers are having numerator is less than denominator always [i.e p<q]
e,g 1/2, 3/4, 5/7, 3/11, 2/7, 4/9, 6/11, 3/10 etc.
The value of every number is less than 1 for always.
b]improper fractions:These numbers are having numerator is greater than denominator always [i.e p>q]
e,g 3/2, 5/4, 7/5, 11/4, 7/2, 9/5, 11/7, 10/3 etc
The value of every number is more than 1 for always.
c]mixed fractions: these numbers are combination of an integer and proper fraction i.e other form of the improper fraction.
e.g 13/4 , 22/3 etc.
The value of every number is more than 1 for always.
Conversion of the improper fraction in to mixed fraction:
Take an improper fraction i.e p/q
Then dived the numerator with denominator identify the quotient and reminder
if the quotient is X and the reminder is y, Take the quotient as complete number, reminder as numerator and denominator of the improper fraction as the denominator of the proper fraction in mixed fraction, finally simply write the mixed fraction equal to the improper fraction given as X y/q
e.g: 7/3 is an improper fraction as 7>3
divide the 7 by 3, we can obtain a quotient as 2 [2x3=6, reminder =7-6=1]
then mixed fraction= 21/3
Conversion of the mixed fraction in to improper fraction:
e.g take a mixed fraction as X y/q
Multiply the complete number with the denominator of it's proper fraction and add the product to the numerator. write the sum as numerator of the proper fraction and denominator is same
E.g 3 2/7 is mixed fraction. Complete number[X]=3, denominator[q]= 7 and numerator[y]=2
simply as above numerator of the improper fraction[p]=Xxq+y=3x7+2=23
denominator is same
Then finally improper fraction=23/7
Decimal fractions:
The proper improper and mixed fractions can be changed to decimal fractions i.e in the form of 10th parts, 100th parts,1000th parts .......os the 1.
e.g 1/2=0.5
2] 1 [0.5 2 can not divide 1 i.e can divide 0 times.
0 reminder is 1, for further division, reminder indicated by 10 part of 1
Fractions:all rational numbers except complete integers are called fractions
these are in the form of p/q.further the numbers are classified as below
a]proper fractions
b]improper fractions
c]mixed fractions
a]proper fractions:These numbers are having numerator is less than denominator always [i.e p<q]
e,g 1/2, 3/4, 5/7, 3/11, 2/7, 4/9, 6/11, 3/10 etc.
The value of every number is less than 1 for always.
b]improper fractions:These numbers are having numerator is greater than denominator always [i.e p>q]
e,g 3/2, 5/4, 7/5, 11/4, 7/2, 9/5, 11/7, 10/3 etc
The value of every number is more than 1 for always.
c]mixed fractions: these numbers are combination of an integer and proper fraction i.e other form of the improper fraction.
e.g 13/4 , 22/3 etc.
The value of every number is more than 1 for always.
Conversion of the improper fraction in to mixed fraction:
Take an improper fraction i.e p/q
Then dived the numerator with denominator identify the quotient and reminder
if the quotient is X and the reminder is y, Take the quotient as complete number, reminder as numerator and denominator of the improper fraction as the denominator of the proper fraction in mixed fraction, finally simply write the mixed fraction equal to the improper fraction given as X y/q
e.g: 7/3 is an improper fraction as 7>3
divide the 7 by 3, we can obtain a quotient as 2 [2x3=6, reminder =7-6=1]
then mixed fraction= 21/3
Conversion of the mixed fraction in to improper fraction:
e.g take a mixed fraction as X y/q
Multiply the complete number with the denominator of it's proper fraction and add the product to the numerator. write the sum as numerator of the proper fraction and denominator is same
E.g 3 2/7 is mixed fraction. Complete number[X]=3, denominator[q]= 7 and numerator[y]=2
simply as above numerator of the improper fraction[p]=Xxq+y=3x7+2=23
denominator is same
Then finally improper fraction=23/7
Decimal fractions:
The proper improper and mixed fractions can be changed to decimal fractions i.e in the form of 10th parts, 100th parts,1000th parts .......os the 1.
e.g 1/2=0.5
2] 1 [0.5 2 can not divide 1 i.e can divide 0 times.
0 reminder is 1, for further division, reminder indicated by 10 part of 1
10 the 2 can divide 5 times.the quotient 5 written after the decimal point.
10
0
10
0
The types of the decimal fractions.
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